Associativity of the singular cap product #
For singular chains and cochains with coefficients in a commutative ring, iterated capping agrees with capping by the cup product:
(x ⌢ α) ⌢ β = x ⌢ (α ⌣ β).
The equality first holds on chains. Both sides evaluate an n-simplex on the same three faces:
the front p-face for α, the following q-face for β, and the back r-face retained as
the resulting chain. It then descends to singular homology and cohomology.
The coefficients are the tensor unit of ModuleCat k; its unitors implement both multiplication
of cochain values and their action on chains. This is the ordinary cap--cup associativity law of
A. Hatcher, Algebraic Topology, Section 3.3.
Cap--cup associativity on singular chains: capping successively with cochains φ and
ψ is capping once with their cup product.
Cap--cup associativity in singular homology: for ordinary cohomology and homology with
coefficients in a commutative ring, capping successively by α and β is capping by
α ⌣ β.